Abstract A subgroup of a group G is said to be hereditarily -permutable with B , if $AB^x = B^xA$ for some element $x \in \langle A, \rangle $ . in every In this paper, we investigate the structure finite all its Schmidt subgroups -permutable.

In this paper we study groups generated by a set X with the property that every two elements in X generate a nilpotent subgroup.

In this paper we will prove analogues of B. Maier’s characterization of weak and strong embeddability of amalgams in N2 [12, 13] for the subvarieties of N2. We will also give analogues of D. Saracino’s characterization of weak and strong amalgamation bases [17], the author’s work on dominions [11] and on amalgamation bases in some varieties of nil-2 groups [8, 9]. Definitions will be recalled b...

Mazur’s theorem states that there are exactly 15 possibilities for the torsion subgroup of an elliptic curve over the rational numbers. We determine how often each of these groups actually occurs. Precisely, if G is one of these 15 groups, we show that the number of elliptic curves up to height X whose torsion subgroup is isomorphic to G is on the order of X, for some number d = d(G) which we c...

The theory of fuzzy sets was inspired by Zadeh [10]. Subsequently, Rosenfeld introduced the concept of a fuzzy subgroup of a group [9]. Fuzzy cosets and fuzzy normal subgroups of a group G have been studied in [3, 5, 8]. In [4], the ring of cosets of a fuzzy ideal was constructed. Let A and B be fuzzy subgroups of G such that B ⊆ A. The purpose of this paper is to introduce the notion of fuzzy ...

Group here always means a locally compact Hausdorff group, subgroup means a closed subgroup. Let G be a group, H a subgroup and G/H the locally compact homogeneous space of left cosets x = xH. We denote by $(G) [®(G/H)] the family of all compact subsets of G [G/H], The group G acts on G/H in a natural way. If X C.G and Y QG/H, write XY for the set of all elements xy, x £ I , j £ Y. Now assume t...

In this paper we recognize the topological type of the group D(R) of C diffeomorphisms of the real line, endowed with the Whitney C-topology (cf. [6]). Besides the whole group D(R) we shall also be interested in its subgroup D c (R) ⊂ D (R) consisting of diffeomorphisms f : R → R with compact support supp(f) = clR{x ∈ R : f(x) 6= x}. For n = 0 the topological group D(R) turns into the homeomorp...

Let p be an odd prime and S a finite p-group. B. Oliver’s conjecture arises from an open problem in the theory of p-local finite groups. It is the claim that a certain characteristic subgroup X(S) of S always contains the Thompson subgroup. In previous work the first two authors and M. Lilienthal recast Oliver’s conjecture as a statement about the representation theory of the factor group S/X(S...

Let X be an arbitrary hyperbolic geodesic metric space and let Γ be a countable subgroup of the isometry group Iso(X) of X. We show that if Γ is non-elementary and weakly acylindrical (this is a weak properness condition) then the second bounded cohomology groups H 2 b (Γ, R), H 2 b (Γ, ℓ p (Γ)) (1 ≤ p < ∞) are infinite dimensional. Our result holds for example for any subgroup of the mapping c...